In this section, we establish an almost sure convergence for the KKT residual \(\nabla \mathcal {L}_t\)  of Algorithm , under standard assumptions. This type of convergence guarantee differs from the convergence in expectation, that is \(\liminf _{t\rightarrow \infty }\mathbb {E}[\Vert \nabla \mathcal {L}_t\Vert ] = 0\) , established in [1]}, [2]}, [3]}, [4]}.
On the other hand, those convergence in expectation results may be reformed to almost sure convergence by applying our following analysis on the \(\ell _1\)  (or \(\ell _2\) ) merit function.
